center of gravity

The point from which the gravitational attraction of a body appears to act.
In a uniform gravitational field
this is coincident with the body's center
of mass. In a distributed mass, the center of gravity (CG) is an appropriately
defined "average location" of its parts. If the mass is a rigid body subject
to the Earth's gravity, then if it is supported at the CG it will stay balanced
and not tilt to any side. In a system subject only to internal forces, the
CG always stays in the same spot; hence the Earth-Moon system rotates around
its mutual CG (not around the Earth's center), and a rocket flies
forward when it ejects a high-speed stream of gas backward. For an expanded
body or collection of particles subject to gravitation, the CG is the point
through which the resultant force of gravity acts, regardless of the orientation
of the body.

Determining the center of gravity

In general, determining the center of gravity is a complicated procedure
because the mass (and weight) may not be uniformly distributed throughout
the object. The general case requires the use of calculus which we will
discuss later. If the mass is uniformly distributed, the problem is greatly
simplified. If the object has a line (or plane) of symmetry, the CG lies
on the line of symmetry. For a solid block of uniform material, the center
of gravity is simply at the average location of the physical dimensions.
(For a rectangular block, 50 × 20 × 10, the center of gravity
is at the point (25, 10, 5).) For a triangle of height h, the CG
is at h/3, and for a semicircle of radius r, the cg is
at (4r/(3π)) where π is ratio of the circumference of the
circle to the diameter. There are tables of the location of the center of
gravity for many simple shapes in math and science books. The tables were
generated by using the equation from calculus shown in the diagram.

For an object of general shape, there is a simple mechanical way to determine
the center of gravity:

If we just balance the object using a string or an edge, the point
at which the object is balanced is the center of gravity. (Just like
balancing a pencil on your finger!)

Another, more complicated way, is a two-step method illustrated above.
In Step 1, you hang the object from any point and you drop a weighted
string from the same point. Draw a line on the object along the string.
For Step 2, repeat the procedure from another point on the object You
now have two lines drawn on the object which intersect. The center of
gravity is the point where the lines intersect. This procedure works
well for irregularly shaped objects that are hard to balance.

If the mass of the object is not uniformly distributed, we must use calculus
to determine center of gravity. The center of gravity can be determined
from:

CG × W = ∫ x dw

where x is the distance from a reference line, dw is an
increment of weight, and W is the total weight of the object. To
evaluate the right side, we have to determine how the weight varies geometrically.
From the weight equation, we know that:

w = m × g

where m is the mass of the object, and g is the gravitational
constant. In turn, the mass m of any object is equal to the density,
ρ (rho), of the object times the volume, V:

m = ρ × V

We can combine the last two equations:

w = g × ρ × V

then

dw = g × ρ × dV

dw = g × ρ(x,y,z)
× dx dy dz

If we have a functional form for the mass distribution, we can solve the
equation for the center of gravity:

CG × W = g × ∫∫∫ x × ρ(x,y,z) dx dy dz

where ∫∫∫ indicates a triple integral over dx, dy,
and dz. If we don't know the functional form of the mass distribution,
we can numerically integrate the equation using a spreadsheet. Divide the
distance into a number of small volume segments and determining the average
value of the weight/volume (density times gravity) over that small segment.
Taking the sum of the average value of the weight/volume times the distance
times the volume segment divided by the weight will produce the center of
gravity.